Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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QPdoll

about 9 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 12 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 23 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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November 25, 2023 at 10:04am

#1060085

I Might Have Overused the Word 'Doom'

Destiny, fate, and doom are roughly synonymous with each other. Only doom, however, conveys the appropriate sense of impending ruination.

‘It’s absolutely guaranteed’: the best and worst case scenarios for sea level rise

I used to think that "doom" was etymologically related to "damn," but this is why we look things up instead of going with our initial guesses.

Not only is dangerous sea level rise “absolutely guaranteed”, but it will keep rising for centuries or millennia even if the world stopped emitting greenhouse gases tomorrow, experts say.

Oi! You're not supposed to tell them that. It makes people wonder why they're bothering to do anything.

Rising seas are one of the most severe consequences of a heating climate that are already being felt.

Okay, but at least we know where that impact will be felt: near sea level. Other consequences can and will (and do) hit anywhere.

To stop the acceleration of sea level rise over the past century, Bamber says, we would have to go back to pre-industrial temperatures.

And the only way that's going to happen is if humanity gets (at least mostly) wiped out. Which, despite the confident claims of people even more pessimistic than I am, climate change probably won't do that by itself.

“The thing about sea level rise is that it is absolutely guaranteed,” Bamber says. “If you warm the planet, sea level is going to go up, period, no caveats. The oceans warm up and the ice melts. It’s an absolute given of global heating.”

Simple solution: build a space elevator. Better yet, build two, one in each of the major oceans. Use them to suck up the excess water and release it into space. As a bonus, Earth will finally get that set of rings it's jealous of Saturn for having.

The impact is hard to gauge because the ocean does not rise at the same speed uniformly, it’s not like a bath. For one thing, Earth is not a perfect sphere; temperatures are also different across the planet, and are affected by ocean currents.

This sort of thing, which I found out about long before reading this article, took me a while to wrap my head around. I mean, sea level is sea level, right? Sure, tides, yeah, but you can average those out and the result should be the same datum for every coast, right?

Wrong.

I think, but I'm not completely sure, that I came to this realization while learning about the Suez Canal. What? Don't look at me like that; I'm a civil engineer and, for once, I was learning about something actually related to my career. Apparently, originally, the Suez Canal flowed one way in summer and the other in winter, due to seasonal differences in sea level between the Mediterranean and Red Seas.

Point is, as the article notes, many different factors contribute to sea level at any given location.

So how bad could things get? Again, it’s hard to predict exactly, but the IPCC has tried its hand at modelling different scenarios for how high sea levels will rise by 2100, based on how well humanity succeeds in mitigating the climate crisis.

I'll make a prediction, too: I predict that, in 2100, if things aren't as bad as the worst-case scenario, some people will gleefully point at the prediction from 2023 and say, "See? Look! Things aren't nearly as bad as they predicted. We did all that for nothing!" And then they'll book themselves a scuba trip to the ruins of Old New Orleans.

“[Where] it’s going to make a huge difference [is] not for us, but for our children’s children. That’s the difficult thing to get your head around.”

And that's why we won't do nearly enough: the people who can actually make a difference only look at quarterly earnings over the next year or so, or the next election in 2-6 years.

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